Small obstacle asymptotics for a 2D semi-linear convex problem

References

• Holland R, Simpson L. Finite-difference analysis of EMP coupling to thin struts and wires. IEEE Trans Electromagn Compat. 1981;2:88–97.
   Google Scholar
• Bérenger J-P. A multiwire formalism for the FDTD method. IEEE Trans Electromagn Compat. 2000;42(3):257–264.
   Web of Science ®Google Scholar
• Railton CJ, Koh BP, Craddock IJ. The treatment of thin wires in the FDTD method using a weighted residuals approach. IEEE Trans Antennas Propag. 2004;52(11):2941–2949.
   Web of Science ®Google Scholar
• Chen Z, Yue X. Numerical homogenization of well singularities in the flow transport through heterogeneous porous media. Multiscale Model Simul. 2003;1(2):260–303.
   Web of Science ®Google Scholar
• Peaceman DW. Interpretation of well-block pressures in numerical reservoir simulations. Soc Pet Eng J. 1978;18(3):183–194.
   Google Scholar
• Van Dyke M. Perturbation methods in fluid mechanics. Stanford (CA): The Parabolic Press; 1975.
   Google Scholar
• Kevorkian J, J ad, Cole D. Perturbation methods in applied mathematics. Vol. 34, Applied mathematical sciences. New York (NY): Springer-Verlag; 1981.
   Google Scholar
• Il’in AM. Matching of asymptotic expansions of solutions of boundary value problems. Vol. 102, Translations of mathematical monographs. Providence (RI): AMS; 1992.
   Google Scholar
• Maz’ya VG, Nazarov SA, Plamenevskiĭ BA. Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vols. 1, 2, Basel: Birkhäuser; 2000. Translated from the original German 1991 edition.
   Google Scholar
• Il’in AM. A boundary value problem for the second order elliptic equation in a domain with a narrow slit 2 domain with a small cavity. Math USSR Sb. 1977;32:227–244.
   Google Scholar
• Maz’ya VG, Nazarov SA, Plamenevskiĭ BA. Asymptotic expansions of eigenvalues of boundary value problems for the Laplace operator in domains with small openings. Izv Akad Nauk SSSR Ser Mat. 1984;48(2):347–371.
   Google Scholar
• Campbell A, Nazarov SA. Une justification de la m\’{e}thode de raccordement des d\’{e}veloppements asymptotiques appliqu\’{ee} \‘{a} un probl\‘{e}me de plaque en flexion. Estimation de la matrice d’impédance. J Math Pures Appl. 1997;76(1):15–54.
   Google Scholar
• Nazarov SA. Asymptotic conditions at a point, selfadjoint extensions of operators, and the method of matched asymptotic expansions. Proceedings of the St. petersburg mathematical society Vol. V. Vol. 193, American mathematical society translations: series 2. Providence (RI): American Mathematical Society; 1999. p. 77–125.
   Google Scholar
• Nazarov SA, Sokołowski J. Asymptotic analysis of shape functionals. J Math Pures Appl. 2003;82(2):125–196.
   Web of Science ®Google Scholar
• Nazarov SA, Sokołowski J. Self-adjoint extensions for the Neumann Laplacian and applications. Acta Math Sin (Engl Ser). 2006;22(3):879–906.
   Web of Science ®Google Scholar
• Cassier M, Hazard C. Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: mathematical justification of the Foldy-Lax model. Wave Motion. 2013;50(1):18–28.
   Web of Science ®Google Scholar
• Amstutz S. Topological sensitivity analysis for some nonlinear PDE systems. J. Math. Pures Appl. 2006;85(4):540–557.
   Web of Science ®Google Scholar
• Iguernane M, Nazarov SA, Roche J-R, et al. Topological derivatives for semilinear elliptic equations. Int J Appl Math Comput Sci. 2009;19(2):191–205.
   Web of Science ®Google Scholar
• Scheid J-F, Sokolowski J, Szulc K. A numerical method for shape and topology optimization for semilinear elliptic equation. 2010 15th International Conference on Methods and Models in Automation and Robotics. 2010.
   Google Scholar
• Drogoul A, Aubert G. The topological gradient method for semi-linear problems and application to edge detection and noise removal. Inverse Prob. Imaging. 2015;10(1):51–86.
   Web of Science ®Google Scholar
• Bonnafé A. Topological asymptotic expansions for a class of quasilinear elliptic equations. Estimates and asymptotic expansions of condenser p-capacities. The anisotropic case of segments [PhD thesis], Toulouse, INSA. 2013.
   Google Scholar
• Amstutz S, Bonnafé A. Topological asymptotic analysis for a class of quasilinear elliptic equations. 2014. hal-00952109. Available from: https://hal.archives-ouvertes.fr/hal-00952109
   Google Scholar
• Kavian O. Introduction à la théorie des points critiques [Introduction to the theory of critical points]. Vol. 13. Mathématiques & applications: Springer-Verlag; 1993.
   Google Scholar
• Grisvard P. Elliptic problems in nonsmooth domains. London: Pitman; 1985.
   Google Scholar
• Nazarov SA, Plamenevskiĭ BA. Elliptic problems in domains with piecewise smooth boundaries. Vol. 13. Berlin, Germany: Expositions in mathematics. De Gruyter; 1994.
   Google Scholar
• Abraham R, Marsden JE, Ratiu T. Manifolds, tensor analysis, and applications, Applied mathematical sciences. 2nd ed. vol. 75. New York (NY): Springer-Verlag; 1988.
   Google Scholar
• Kondratiev VA. Boundary-value problems for elliptic equations in domains with conical or angular points. Trans Moscow Math Soc. 1967;16:227–313.
   Google Scholar
• Nédélec J-C. Acoustic and electromagnetic equations: integral representations for harmonic problems. Vol. 144. Springer; 2001.
   Google Scholar
• Landkof NS. Foundations of modern potential theory. New York (NY): Springer-Verlag; 1972. Translated from the Russian by A.P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.
   Google Scholar
• Pólya G, Szegö G. Isoperimetric inequalities in mathematical physics. Annals of mathematics studies. No. 27, Princeton (NJ): Princeton University Press; 1951.
   Google Scholar
• Chesnel L, Claeys X. A numerical approach for the poisson equation in a planar domain with a small inclusion. BIT. 2016;56:1237–1256. doi:10.1007/s10543-016-0615-z.
   Web of Science ®Google Scholar
• Ransford T. Computation of logarithmic capacity. Comput Meth Funct Theor. 2011;10(2):555–578.
   Web of Science ®Google Scholar